In general, by a fake complex projective \(n\)-space we mean
a smooth manifold \(F_{\mathbb {C}P^{n}}\)  homotopy equivalent to the complex
projective space \(\mathbb {C}P^{n}\) , \(n\ge 2\) . For the classifications of
such manifolds, see [1]} and [2]}. In
particular, if we let \(p:E\rightarrow F_{\mathbb {C}P^{n}}\)  be the oriented
circle bundle on \(F_{\mathbb {C}P^{n}}\)  whose Euler class generates the group
\(H^{2}(F_{\mathbb {C}P^{n}})=\mathbb {Z}\) , then \(E\)  is a homotopy \((2n+1)\) -spheres. In addtion, since the second Stiefel-Whitney class is a homotopy
invariant of smooth manifolds, we have \(w_{2}(F_{\mathbb {C}P^{n}})\ne 0\)  if
and only if \(n\)  even. Thus, Theorem B implies that, for any fake complex
projective space \(F_{\mathbb {C}P^{n}}\)  and any \(N\in \mathcal {T}_{2n}\) , the
(topological) manifold \(\Sigma _{i}N\)  admits a smooth structure and a
regular circle action, whose quotient is diffeomorphic to \(F_{\mathbb {C}P^{n}}\#N\) , where \(n=i{mod}2\) .\(\square \)
